Verstappen has been rewriting the history book this season with the dominant RB19, having been undefeated since team-mate Sergio Perez's win in Baku.
His win in last weekend's Italian Grand Prix was his 10th victory in a row, beating the all-time record he shared with Sebastian Vettel and Alberto Ascari, while Red Bull's streak of 15 wins is also unprecedented.
Verstappen has regularly played down the significance of his achievements and said he isn't overly concerned by chasing records.
But behind the scenes his team feel that the seemingly unflappable Dutchman did care more about the win record than he let on in public.
"I wouldn’t say nervous, but I would say you could definitely feel that his focus was razor sharp, more than usual. You could see that it definitely meant something to him," team principal Horner said.
"To get that 1-2 finish and for Max to be on his own as the most consecutive F1 wins at 10 victories is quite outstanding."
Red Bull advisor Helmut Marko did feel some nerves with Verstappen before the race as it was "important" to grab the record during a uniquely dominant campaign.
"For Max, this winning streak means a lot," Marko said. "For me personally, getting the 100th win as a team was even more special than this, but it is fantastic for Max.
"Max was a little bit nervous to get it done here as well, for him it was really important I believe to break this record.
"Now he has it, so I think that is a bit more relaxing for the races that are coming."
Verstappen's win record was met with indifference by Horner's Mercedes counterpart Toto Wolff, who said it “is for Wikipedia and nobody reads that anyway”.
It followed comments by Lewis Hamilton unfavourably comparing the strength of Verstappen's team-mates to his own.
But Horner said he is keen to move on from a recent flare-up of his rivalry with Wolff and wanted to focus on Red Bull's own achievements.
"Look, I don't want to get drawn into commenting on that," he said.
"Max is driving at an unbelievable level and I don't think there's anybody in the world right now that could beat Max Verstappen in this car, that's for sure.
"I think you have to recognise and applaud what Max is doing at the moment, we shouldn't detract from that in any way.
"In sport, very rarely things like this happen, and it's a golden moment for him and certainly a golden moment for the team."
Additional reporting by Ronald Vording
